Equal Sets

What Does It Mean for Two Sets to Be Equal?

Two sets are equal when they contain exactly the same elements — no more, no fewer. The order in which the elements are listed does not matter, and repeating an element does not change the set. If every element of $A$ is also an element of $B$, and every element of $B$ is also an element of $A$, then $A = B$. If this condition fails, we write $A \neq B$.

Examples That Clarify the Idea

Example 1.

$A = \{1, 2, 3, 4\}$ and $B = \{3, 1, 4, 2\}$.

Even though the elements are listed in a different order, every element of $A$ appears in $B$ and vice versa. Hence $A = B$.

Example 2.

Let $A$ be the set of prime numbers less than $6$: $A = \{2, 3, 5\}$.

Let $P$ be the set of prime factors of $30$: $30 = 2 \times 3 \times 5$, so $P = \{2, 3, 5\}$.

Since the two sets are identical, $A = P$.

Example 3 (repetition of elements).

$A = \{1, 2, 3\}$ and $B = \{2, 2, 1, 3, 3\}$.

Repetition does not create new elements. The set $B$ still contains only $1$, $2$, and $3$. Therefore $A = B$. This is why we never write repeated elements when describing a set — it is redundant.

Key Points to Remember

• Two sets are equal if and only if they have exactly the same elements.
• The order of listing elements is irrelevant.
• Repeating an element does not change the set.
• To check equality, verify that every element of the first set is in the second, and every element of the second is in the first. …